**1. Introduction**

As is known, the formation of microdroplets and their penetration into the structures of modifying coatings are a significant problem, typical of the physical vapour deposition (PVD) method [1–5]. Technologies and equipment—from simple arrays to complex magnetic traps—are being actively developed and used to significantly reduce the number of microdroplets [6–12]. At the same time, the process of the formation of microdroplets and their influence on the coating structure are being studied. In particular, it has been found that the number and mean size of microdroplets increase simultaneously with an increase in the discharge current, the gas pressure, and temperature of the cathode surface [13–16]. In [13,16,17], it is noted that the microdroplet shapes vary, and microdroplets can be classified into three groups, defined as sphere-, lens-, and tear-shaped microdroplets. The number of microdroplets decreases with an increase in the temperature of the substrate [18]. The processes of microdroplet formation and penetration into the coating structure has been studied mainly by using mathematical modeling methods. In several papers, the studies were focused on the mechanism of

microdroplet deformation upon hitting a substrate in various deposition conditions. In [19], the cooling of a liquid microdroplet is modeled using the laws of hydrodynamics and assuming an ideal interphase thermal contact. The Lagrangian concept and the hydrodynamic model of Fukai et al. [20] were used to simulate the heat transfer in a microdroplet and a substrate. Wadvogel et al. [21,22] expanded the specified model, taking into account the solidification process and the incomplete interphase thermal contact. Several studies separately consider the role of incomplete thermal contact between a substrate and a microdroplet, which is an essential parameter in the process of heat transfer. In particular, Liu et al. [23] suggest that no ideal thermal contact between a microdroplet and a substrate can be achieved because of the roughness of the substrate surface, the surface tension of the microdroplet, the presence of surface impurities, and gas entrapment. Heat transfer in the real area of contact between the droplet and the substrate surface occurs both due to thermal conductivity and, to some extent, radiation [24]. For microdroplets of molten lead and similar materials used for brazing, the incomplete thermal contact was investigated in [25–27] through experiments. In particular, it was found that the incomplete thermal contact affects the microdroplet shape, and their height relative to the substrate can change by 20% due to changes in the thermal contact resistance. Some values of the interphase heat transfer coefficients for specific pairs of materials were determined both by calculations and experiments in [28]. It was found that the interphase heat transfer coefficient noticeably influences the microdroplet solidification process. In [29], the study focused on the modeling of a liquid microdroplet with the diameter of 80 μm that moved with a velocity of 5 m/s and hit a flat solid surface. Given that the conditions under consideration differ from the coating deposition conditions, this model can be useful, since the general nature of the physical conditions (sizes of the microdroplet, its movement velocity, and penetration of the liquid microdroplet into the solid surface) correlates well with the challenge under consideration. In [29], the investigation also takes into account the possible variance of the heat transfer coefficient over time. In [30], the studies simulate a hollow microdroplet, consisting of a liquid shell enclosing a gas cavity and hitting a solid flat surface. The studies demonstrate that the deformation nature of a hollow microdroplet, upon impact with a solid flat surface, differs significantly from the deformation of a filled microdroplet. In [31], the studies simulate the influence of parameters—such as the microdroplet velocity on impact, the relative distances between two successive microdroplets of molten metal, the substrate temperature, and the microdroplet sizes—on the morphology of the final shape of a microdroplet solidified after the impact. The mechanisms of the microdroplet formation are also considered in detail in [1,32,33]. All the simulation results presented indicate that upon contact of a liquid or highly ductile microdroplet with the deposition surface, the microdroplet is subjected to deformation, and it forms a typical lens-shaped structure, sometimes with a torus-shaped formation along the microdroplet boundaries. All the above papers consider the microdroplet as a liquid or highly ductile substance which does not change during the process of movement from a source to the deposition surface. At the same time, the monitoring proves that a coating structure can include not only lens-shaped microdroplets, but also microdroplets of an almost ideal spherical shape. Such a microdroplet could form only during the cooling until it contacts the deposition surface, otherwise, the microdroplet would have been inevitably deformed.

In [34], Anders considered, in sufficient detail, the issues concerning the formation of microdroplets, their movement from a cathode to a deposition surface, and their effect on the coating structure. In particular, the paper notes that microdroplets can move along trajectories that are different from a simple rectilinear trajectory, and they can repeatedly bounce off the deposition surface. Meanwhile, only the trajectories of large (macro-) droplets of tens of μm can be visually traced. In [34], it is also noted that upon hitting a deposition surface, most of the microdroplets take a donut-like shape; however, some microdroplets (according to the author, the microdroplets with the diameter less than 0.5 μm) solidify before they hit the deposition surface and take a spherical shape. A microdroplet moves in the nitrogen environment, and accordingly, a nitride shell is formed on the microdroplet surface. This is one of the possible reasons for the formation of a solid microdroplet with a regular spherical shape. For example, while moving in the nitrogen environment, a liquid titanium microdroplet can

form a shell of TiN, since the melting point of titanium nitride (2930 ◦C) is much higher than that of pure titanium (1668 ◦C). As a result, the microdroplet reaching the deposition surface can have a solid nitride shell.

Multicomponent coatings based on nitrides of metals such as Ti, Al, Cr, Mo, Nb, and Zr were analyzed in a large number of studies. In particular, a number of studies have focused on the properties of the (Zr,Nb)N coating, in which a cubic fcc-(Zr,Nb)N mixed crystal phase has been detected as the dominant one [35–38]. The (Ti,Zr)N coating has also been investigated in a number of papers [39–47]. In [39–41], the presence of two cubic phases of fcc-TiN and fcc-ZrN has been detected. Meanwhile, the phase of (Ti,Zr)N (solid solution of Zr in fcc-TiN) is dominating [42–45]. It should also be noted that the hardness and heat resistance of the (Ti,Zr)N coating are significantly higher than those of TiN or ZrN [43,46,47]. The (Ti,Mo)N and (Ti,Cr)N coatings have been studied in [48–52]. In particular, in [48,49], the studies have revealed that the phase of fcc-TiN with Mo dissolved in it dominates in the (Ti,Mo)N coating. An important advantage of the coatings containing Cr and Mo, is their ability to form the dense oxide films (based on Magneli phases) at elevated temperatures, and these films perform tribological and protective functions [50–52]. The studies focused on the (Cr,Mo)N coating have detected the formation of a solid solution of Mo in fcc-CrN [53,54]. In [55], the formation of the tribological films of MoO3 and Cr2O3 has been revealed, with the oxide of MoO3 having a key effect on the reduction in friction. The properties of four-component coatings based on metals of the indicated elements have also been investigated. In particular, the study focused on the properties of the (Ti,Al,Zr)N coating has found the presence of a multiphase structure with a dominant phase based on fcc-TiN with dissolved Al and Zr, but phases of Zr3A1N, Ti3A1N, and A1N have also been detected [56,57]. The coating of (Ti,Al,Mo)N has also been studied in [58–62], MoNAl and Mo in fcc-TiN with the presence of an insignificant amount of MoN phase. The results of the studies performed demonstrate that the multicomponent coatings based on a solid solution of such elements as Cr, Mo, Al, Nb, and Zr in fcc-TiN can significantly increase the hardness, wear and heat resistance.

This study deals with the formation of a microdroplet phase in the coatings of Ti–TiN–(Ti,Cr,Mo,Al)N and Ti–TiN–(Ti,Al,Nb,Zr)N. Earlier, other coatings with similar compositions, including Cr,Mo–(Cr,Mo,Zr,Nb)N–(Cr,Mo,Zr,Nb,Al)N, Cr,Mo–(Cr,Mo)N–(Cr,Mo,Al)N, and Zr,Nb–(Zr,Nb)N–(Zr,Nb,Al)N, had been considered, and they proved to have good mechanical properties, while cutting tools with these coatings demonstrated good cutting properties [60]. Meanwhile, the current study is focused on microdroplets embedded into the structure of the coatings under consideration. It should be noted that the investigation of microdroplets in their dynamic state (while they are moving from a cathode to the deposition surface) is a hard-hitting challenge due to their microscopic sizes, stochastic trajectory of motion, and difficulties in observing in conditions of plasma flow. Accordingly, there are two possible ways to study the above objects, that is, to study solidified microdroplets in the coating structure and to apply methods of mathematical modeling. It should be noted that the development of an adequate model for the motion of a microdroplet is another hard-hitting challenge, taking into account a number of stochastic factors affecting the process.
